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Description

This study calculates and displays a Butterworth Filter for the data given by the Input Data Input. This study is an ACSIL implementation of the Indicators given in Figures 13.4, 13.5, 13.6, and 13.7 of the book Cybernetic Analysis for Stocks and Futures by John Ehlers.

Let \(X\) be a random variable denoting the Input Data, and let \(X_t\) be the value of \(X\) at Index \(t\). Let the Length Input be denoted as \(n\).

The Poles Input determines whether the 2-Pole or 3-Pole Butterworth Filter is computed. The values of these two filters are denoted respectively as \(BWF^{(2)}_t(X,n)\) and \(BWF^{(3)}_t(X,n)\), and they are computed as follows.

2-Pole Butterworth Filter:

where the coefficients \(k^{(1)}(n)\), \(k^{(2)}(n)\), and \(k^{(3)}(n)\) are given as follows.

• \(\displaystyle{k^{(1)}(n) = \frac{1}{4}\left(1 - 2\exp\left(-\frac{\sqrt{2}\pi}{n}\right)\cos\left(\frac{\sqrt{2}\pi}{n}\right) + \mathrm{exp}\left(-\frac{2\sqrt{2}\pi}{n}\right)\right)}\)
• \(\displaystyle{k^{(2)}(n) = 2\exp\left(-\frac{\sqrt{2}\pi}{n}\right)\cos\left(\frac{\sqrt{2}\pi}{n}\right)}\)
• \(\displaystyle{k^{(3)}(n) = -\exp\left(-\frac{2\sqrt{2}\pi}{n}\right)}\)

3-Pole Butterworth Filter:

where the coefficients \(k^{(1)}(n)\), \(k^{(2)}(n)\), \(k^{(3)}(n)\), and \(k^{(4)}(n)\) are given as follows.

• \(\displaystyle{k^{(1)}(n) = \frac{1}{8}\left(1 - 2\exp\left(-\frac{\pi}{n}\right)\cos\left(\frac{1.738\pi}{n}\right) + \exp\left(-\frac{2\pi}{n}\right)\right)\left(1 - \exp\left(-\frac{2\pi}{n}\right)\right)}\)
• \(\displaystyle{k^{(2)}(n) = 2\exp\left(-\frac{\pi}{n}\right)\cos\left(\frac{1.738\pi}{n}\right) + \exp\left(-\frac{2\pi}{n}\right)}\)
• \(\displaystyle{k^{(3)}(n) = -\exp\left(-\frac{2\pi}{n}\right) - 2\exp\left(-\frac{3\pi}{n}\right)\cos\left(\frac{1.738\pi}{n}\right)}\)
• \(\displaystyle{k^{(4)}(n) = \exp\left(-\frac{4\pi}{n}\right)}\)

For an explanation of the Exponential Function \(\exp()\), refer to our description here.